Gmunu : Toward multigrid based Einstein field equations CHEONG, - PowerPoint PPT Presentation

Gmunu : Multigrid methods for solving Einstein Gmunu : Toward multigrid based Einstein field equations CHEONG, Chi-Kit field equations solver for (Patrick) Tjonnie G. F. LI, general-relativistic hydrodynamics L. M. LIN simulations Introduction Methods TAUP2019 Results Covergence properties Code test Conclusions CHEONG, Chi-Kit (Patrick) Backup slides Tjonnie G. F . LI, L. M. LIN Department of Physics The Chinese University of Hong Kong Sep 12 2019

Gmunu : Multigrid Outline methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI, Introduction L. M. LIN Introduction Methods Methods Results Covergence properties Results Code test Conclusions Covergence properties Backup slides Code test Conclusions Backup slides

Gmunu : Multigrid Introduction methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI, L. M. LIN ◮ Implementation details of Gmunu (P . C. K. CHEONG Introduction et al., in prep.) Methods ◮ Application of Gmunu (H. Y. NG et al.,Talk on Results Covergence properties Tuesday, in prep.) Code test ◮ Why numerical simulations Conclusions Backup slides ◮ Introduction of CCSNe and its gravitational waves ◮ Develop a code for: ◮ Core-collapse supernovae ◮ Isolated Neutron stars

Gmunu : Multigrid Introduction methods for solving Einstein field equations CHEONG, Chi-Kit Relativistic (Case GR) (Patrick) F Newtonian with modified Tjonnie G. F. LI, F TOV potential and rotation L. M. LIN corrections (Case Arot) Introduction Methods H 1 H 1 H 2 Results H 2 Covergence properties Code test Conclusions Backup slides 1 2 3 4 5 6 7 8 9 f [kHz] B. Muller, et al. 2008 General relativistic simulations is needed!

Gmunu : Multigrid Introduction methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI, Different ways to solve Einstein equations L. M. LIN ◮ “free-evolution” formulations Introduction ◮ constrained formulation Methods Results ◮ elliptic sector and hyperbolic sector Covergence properties Code test Conformally fatness approximation (CFC) is applied Conclusions successfully in various astrophysical problems even in Backup slides spherical-polar coordinate However... ◮ high computational cost for elliptic equations ◮ on the fly simulation?

Gmunu : Multigrid Conformally flatness approximation methods for solving Einstein CFC approximation field equations − α 2 + β i β i CHEONG, Chi-Kit  β 1 β 2 β 3  (Patrick) Tjonnie G. F. LI, ϕ 4 β 1 0 0 L. M. LIN   g µν =  ϕ 4 r 2  β 2 0 0   Introduction ϕ 4 r 2 sin 2 θ β 3 0 0 Methods Results 5 100 Covergence properties ρ max (10 14 g cm − 3 ) A1B3G3 A1B3G3 50 4 Code test h + R (cm) 0 -50 3 Conclusions -100 2 -150 Backup slides -200 50 ρ max (10 13 g cm − 3 ) 6 A2B4G1 A2B4G1 0 h + R (cm) 4 -50 full GR full GR -100 2 CFC CFC -150 5 ρ max (10 14 g cm − 3 ) 30 A1B3G5 A1B3G5 h + R (cm) 20 4 10 0 3 -10 2 -20 -5 0 5 10 15 -5 0 5 10 15 t - t bounce (ms) t - t bounce (ms) Ott, et al. 2006

Gmunu : Multigrid xCFC scheme methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI, Cordero-Carrion et al. (2009) L. M. LIN ∆ X i + 1 ∇ i � ∇ j X j � Introduction = 8 π ˜ ˜ ˜ ˜ S i 3 Methods ∇ j X i − 2 Results ∇ i X j + ˜ A ij ≈ ˜ ˜ ∇ k X k f ij ˜ Covergence properties 3 Code test Eψ − 1 − 1 Conclusions A kl ˜ ∆ ψ = − 2 π ˜ ˜ 8 f ik f jl ˜ A ij ψ − 7 Backup slides � ψ − 2 + 7 � � � A kl ˜ ˜ E + 2 ˜ ˜ 8 f ik f jl ˜ A ij ψ − 8 ∆( αψ ) = ( αψ ) 2 π S ∆ β i + 1 ∇ i � ∇ j β j � = 16 παψ − 6 f ij ˜ A ij ˜ ˜ ˜ ˜ S i + 2 ˜ αψ − 6 � � ∇ j 3

Gmunu : Multigrid Boundary condiditon methods for solving Einstein field equations CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI, L. M. LIN Robin B. C. Introduction Methods ∂ψ r max = 1 − ψ � Results � ∂r r Covergence properties � Code test ∂α r max = 1 − α � Conclusions � ∂r � r Backup slides β i � r max = 0 � �

Gmunu : Multigrid Code features methods for solving Einstein field equations Gmunu ( G eneral-relativistic mu ltigrid nu merical Einstein CHEONG, Chi-Kit solver) (Patrick) Tjonnie G. F. LI, ◮ 3+1 in Spherical polar coordinates (1,2,3-D) L. M. LIN ◮ Hydro: high-resolution shock-capturing (HRSC) Introduction methods Methods ◮ Reconstruction method: Piecewise-Constant (PC), Results Covergence properties TVD, (5-th order)WENO, MP5 Code test ◮ Riemann solver: HLL, HLLE, Marquina Conclusions ◮ Time update: RK3 methods Backup slides ◮ Conserved variables to Primitive variables: Regula-Falsi method ◮ Conformally flatness condition(CFC) metric evolution: ◮ extended CFC (xCFC) scheme (Cordero-Carrion et al. (2009)) ◮ Multigrid solver for the elliptic non-linear coupled equations

Gmunu : Multigrid Key features of our MG solver methods for solving Einstein field equations ◮ Nonlinear FAS (Full Approximation Storage) CHEONG, Chi-Kit (Patrick) algorithm Tjonnie G. F. LI, L. M. LIN ◮ Cycles options: V, W and F Introduction ◮ Cell-Centered discretization (no grid communication Methods is needed) Results ◮ Smoother: Red-Black Gauss-Seidel relaxation Covergence properties Code test ◮ Prolongation: bi-linear Conclusions Backup slides ◮ Restriction: piecewise constant

Gmunu : Multigrid Models methods for solving Einstein field equations Table: Γ = 2 , K = 100 , ( c = G = M sun = 1) . CHEONG, Chi-Kit (Patrick) Tjonnie G. F. LI, ρ c [10 − 3 ] Ω [10 − 2 ] Model L. M. LIN BU0 1 . 28 0 . 000 Introduction SU 8 . 00 0 . 000 Methods 1 . 28 1 . 509 BU2 Results BU8 1 . 28 2 . 633 Covergence properties Code test Conclusions Backup slides

Gmunu : Multigrid Convergence properties methods for solving Einstein ◮ solving the α of BU8 with initial guess: α = 1 field equations ◮ n r × n θ = 640 × 64 CHEONG, Chi-Kit (Patrick) ◮ Gauss-Seidel ( ∼ 5 . 3 × 10 5 ) to MG ( ∼ 50 ) Tjonnie G. F. LI, L. M. LIN ◮ O (minus) → O (ms) Introduction Methods Results 10 2 Covergence properties Code test Conclusions 10 0 Backup slides L 1 norm 10 − 2 10 − 4 V 1 V 2 10 − 6 V 3 V 4 V 5 10 − 8 V 6 0 10 20 30 40 50 60 Number of iterations

Gmunu : Multigrid Model: BU0 methods for solving Einstein ◮ n r × n θ = 640 × 64 field equations CHEONG, Chi-Kit 0 , π ◮ r = [0 , 30] , θ = � � (Patrick) 2 Tjonnie G. F. LI, L. M. LIN 5 . 0 [ ρ c ( t ) /ρ c (0) − 1] × 10 4 2 . 5 Introduction 0 . 0 Methods − 2 . 5 Results Covergence properties − 5 . 0 Code test − 7 . 5 Conclusions − 10 . 0 Backup slides 0 2 4 6 8 10 12 t (ms) FFT of v r ( t ) and v θ ( t ) FFT( v r ) 0 . 06 FFT( v θ ) F 0 . 04 H 1 H 2 H 3 0 . 02 0 . 00 1000 2000 3000 4000 5000 6000 7000 8000 f (Hz)

Gmunu : Multigrid Model: SU methods for solving Einstein field equations Migration test: unstable -> stable NS CHEONG, Chi-Kit (Patrick) ◮ n r × n θ = 1024 × 1 (1D simulation) Tjonnie G. F. LI, L. M. LIN ◮ r = [0 , 34] Introduction Methods 1 . 0 Results Covergence properties 0 . 8 Code test Conclusions ρ c ( t ) /ρ c (0) Backup slides 0 . 6 0 . 4 0 . 2 0 . 0 0 2 4 6 8 t (ms)

Gmunu : Multigrid Model: BU2 methods for solving Einstein ◮ n r × n θ = 640 × 64 field equations CHEONG, Chi-Kit 0 , π ◮ r = [0 , 30] , θ = � � (Patrick) 2 Tjonnie G. F. LI, L. M. LIN 5 . 0 [ ρ c ( t ) /ρ c (0) − 1] × 10 4 2 . 5 Introduction 0 . 0 Methods − 2 . 5 Results Covergence properties − 5 . 0 Code test − 7 . 5 Conclusions − 10 . 0 Backup slides 0 . 0 2 . 5 5 . 0 7 . 5 10 . 0 12 . 5 15 . 0 17 . 5 20 . 0 t (ms) 0 . 08 FFT of v r ( t ) and v θ ( t ) FFT( v r ) FFT( v θ ) 0 . 06 F H 1 0 . 04 2 f 2 p 1 0 . 02 i 2 0 . 00 0 1000 2000 3000 4000 5000 6000 f (Hz)

Gmunu : Multigrid Model: BU2 methods for solving Einstein ◮ n r × n θ = 640 × 64 field equations CHEONG, Chi-Kit 0 , π ◮ r = [0 , 30] , θ = � � (Patrick) 2 Tjonnie G. F. LI, L. M. LIN T max = 20 ms Introduction 1 . 0 equatorial initial Methods equatorial Results 0 . 8 pole initial Covergence properties Code test pole Conclusions 0 . 6 Backup slides ρ/ρ c 0 . 4 0 . 2 0 . 0 0 5 10 15 20 r (km)

Gmunu : Multigrid Model: BU2 methods for solving Einstein ◮ n r × n θ = 640 × 64 field equations CHEONG, Chi-Kit 0 , π ◮ r = [0 , 30] , θ = � � (Patrick) 2 Tjonnie G. F. LI, L. M. LIN T max = 20 ms Introduction 1 . 0 equatorial initial Methods equatorial Results 0 . 8 Covergence properties � Code test v φ v φ Conclusions �� 0 . 6 Backup slides v φ v φ / max 0 . 4 � 0 . 2 0 . 0 0 5 10 15 20 r (km)

Gmunu : Multigrid Model: BU8 methods for solving Einstein ◮ n r × n θ = 640 × 64 field equations CHEONG, Chi-Kit 0 , π ◮ r = [0 , 30] , θ = � � (Patrick) 2 Tjonnie G. F. LI, L. M. LIN 2 [ ρ c ( t ) /ρ c (0) − 1] × 10 4 0 Introduction − 2 Methods − 4 Results Covergence properties − 6 Code test − 8 Conclusions − 10 Backup slides 0 . 0 2 . 5 5 . 0 7 . 5 10 . 0 12 . 5 15 . 0 17 . 5 20 . 0 t (ms) 0 . 4 FFT of v r ( t ) and v θ ( t ) FFT( v r ) FFT( v θ ) 0 . 3 F H 1 0 . 2 2 f 0 . 1 0 . 0 0 1000 2000 3000 4000 5000 6000 f (Hz)